Check the solution to "Determine the following integrals. y(y2+1)54dy"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

dedica66em

Answered question

2021-12-13

Determine the following integrals.

\frac{y}{{(y^{2} + 1)}^{\frac{5}{4}}} d y

Answer & Explanation

Juan Spiller

Beginner2021-12-14Added 38 answers

Step 1
Our Aim is to evaluate the integral given below:−
$\int \frac{y}{{(y^{2} + 1)}^{\frac{5}{4}}} d y$ ...(i)
Step 2
Consider the integral given by equation-(i), we have:−
$\int \frac{y}{{(y^{2} + 1)}^{\frac{5}{4}}} d y - (i)$
For the integrand $\frac{y}{{(y^{2} + 1)}^{\frac{5}{4}}}$ , we will substitute $t = y^{2} + 1$ and dt=2ydy
$= \frac{1}{2} \int \frac{1}{t^{\frac{5}{4}}} d t$
Since, the integral of $\frac{1}{t^{\frac{5}{4}}} i s \frac{- 4}{\sqrt[4]{t}}$
$= \frac{- 2}{\sqrt[4]{t}} + c o n s t a n t$
Answer $= \frac{- 2}{\sqrt[4]{y^{2} + 1}} + c o n s t a n t$

servidopolisxv

Beginner2021-12-15Added 27 answers

Given:
$\int \frac{y}{{(y^{2} + 1)}^{\frac{5}{4}}} d y$
$\int \frac{1}{2 t^{\frac{5}{4}}} d t$
Use properties of integrals
$\frac{1}{2} * \int \frac{1}{t^{\frac{5}{4}}} d t$
$\frac{1}{2} * (- \frac{4}{\sqrt[4]{t}})$
Substitute back
$\frac{1}{2} * (- \frac{4}{\sqrt[4]{y^{2} + 1}})$
$- \frac{2}{\sqrt[4]{y^{2} + 1}}$
Result:
$- \frac{2}{\sqrt[4]{y^{2} + 1}} + C$

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